Properties

Label 6.6.300125.1-71.6-b1
Base field 6.6.300125.1
Conductor norm \( 71 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field 6.6.300125.1

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 7, 2, -7, -1, 1]))
 
gp: K = nfinit(Polrev([-1, -2, 7, 2, -7, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(-3a^{5}+a^{4}+22a^{3}+8a^{2}-17a-3\right){x}{y}+\left(-6a^{5}+a^{4}+44a^{3}+23a^{2}-29a-10\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+a^{2}+a-2\right){x}^{2}+\left(-14a^{5}+3a^{4}+102a^{3}+49a^{2}-67a-19\right){x}-4a^{5}+a^{4}+29a^{3}+14a^{2}-20a-6\)
sage: E = EllipticCurve([K([-3,-17,8,22,1,-3]),K([-2,1,1,-6,-1,1]),K([-10,-29,23,44,1,-6]),K([-19,-67,49,102,3,-14]),K([-6,-20,14,29,1,-4])])
 
gp: E = ellinit([Polrev([-3,-17,8,22,1,-3]),Polrev([-2,1,1,-6,-1,1]),Polrev([-10,-29,23,44,1,-6]),Polrev([-19,-67,49,102,3,-14]),Polrev([-6,-20,14,29,1,-4])], K);
 
magma: E := EllipticCurve([K![-3,-17,8,22,1,-3],K![-2,1,1,-6,-1,1],K![-10,-29,23,44,1,-6],K![-19,-67,49,102,3,-14],K![-6,-20,14,29,1,-4]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^4+a^3+6a^2-a-4)\) = \((-a^4+a^3+6a^2-a-4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 71 \) = \(71\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^4-a^3-6a^2+a+4)\) = \((-a^4+a^3+6a^2-a-4)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -71 \) = \(-71\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1584785}{71} a^{5} - \frac{365527}{71} a^{4} - \frac{11832625}{71} a^{3} - \frac{4916339}{71} a^{2} + \frac{8890196}{71} a + \frac{1422813}{71} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(-a^{5} + a^{4} + 6 a^{3} - 3 a : 2 a^{5} - 16 a^{3} - 7 a^{2} + 11 a + 3 : 1\right)$
Height \(0.0030387827717437604474034092071240670275\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-3 a^{5} + a^{4} + 21 a^{3} + 9 a^{2} - 12 a - 4 : 1 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.0030387827717437604474034092071240670275 \)
Period: \( 267854.18856635194283505339506806078638 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 2.22863 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((-a^4+a^3+6a^2-a-4)\) \(71\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 71.6-b consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.